Difference between revisions of "009B Sample Midterm 1, Problem 2"

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<span class="exam">Find the average value of the function on the given interval.
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<span class="exam"> Otis Taylor plots the price per share of a stock that he owns as a function of time
  
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::<math>f(x)=2x^3(1+x^2)^4,~~~[0,2]</math>
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<span class="exam">and finds that it can be approximated by the function
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::::::<math>s(t)=t(25-5t)+18</math>
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<span class="exam">where <math>t</math> is the time (in years) since the stock was purchased.
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<span class="exam">Find the average price of the stock over the first five years.
  
  
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|Using the formula given in Foundations, we have:
 
|Using the formula given in Foundations, we have:
 
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| &nbsp; &nbsp;<math style="vertical-align: 0px">f_{\text{avg}}=\frac{1}{2-0}\int_0^2 2x^3(1+x^2)^4~dx=\int_0^2 x^3(1+x^2)^4~dx.</math>  
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| &nbsp; &nbsp;<math style="vertical-align: 0px">f_{\text{avg}}=</math>  
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now, we use <math>u</math>-substitution. Let <math style="vertical-align: -2px">u=1+x^2</math>. Then, <math style="vertical-align: 0px">du=2x dx</math> and <math style="vertical-align: -13px">\frac{du}{2}=xdx</math>. Also, <math style="vertical-align: 0px">x^2=u-1</math>.
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|We need to change the bounds on the integral. We have <math style="vertical-align: -4px">u_1=1+0^2=1</math> and <math style="vertical-align: -3px">u_2=1+2^2=5</math>.
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|So, the integral becomes <math style="vertical-align: -19px">f_{\text{avg}}=\int_0^2 x\cdot x^2 (1+x^2)^4~dx=\frac{1}{2}\int_1^5(u-1)u^4~du=\frac{1}{2}\int_1^5(u^5-u^4)~du</math>.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 3: &nbsp;
 
!Step 3: &nbsp;
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|We integrate to get
 
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| &nbsp; &nbsp; <math>f_{\text{avg}}=\left.\frac{u^6}{12}-\frac{u^5}{10}\right|_{1}^5=\left.u^5\bigg(\frac{u}{12}-\frac{1}{10}\bigg)\right|_{1}^5.</math>
 
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!Step 4: &nbsp;
 
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|We evaluate to get
 
 
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| &nbsp; &nbsp; <math style="vertical-align: -20px">f_{\text{avg}}=5^5\bigg(\frac{5}{12}-\frac{1}{10}\bigg)-1^5\bigg(\frac{1}{12}-\frac{1}{10}\bigg)=3125\bigg(\frac{19}{60}\bigg)-\frac{-1}{60}=\frac{59376}{60}=\frac{4948}{5}</math>.
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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| &nbsp; &nbsp; <math>\frac{4948}{5}</math>
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| &nbsp; &nbsp; <math></math>
 
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[[009B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 09:14, 5 February 2017

Otis Taylor plots the price per share of a stock that he owns as a function of time

and finds that it can be approximated by the function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t)=t(25-5t)+18}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} is the time (in years) since the stock was purchased.

Find the average price of the stock over the first five years.


Foundations:  
The average value of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} on an interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)~dx} .

Solution:

Step 1:  
Using the formula given in Foundations, we have:
   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\text{avg}}=}
Step 2:  
Step 3:  
Final Answer:  
   

Return to Sample Exam

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